If possible,let √p be a rational number.
also a and b is rational.
then,√p = a/b
on squaring both sides,we get,
(√p)²= a²/b²
⇒ p = a²/b²
⇒ b² = a²/p [p divides a² so,p divides a]
Let a= pr for some integer r
⇒ b² = (pr)²/p
⇒ b² = p²r²/p
⇒ b² = pr²
⇒ r² = b²/p [p divides b² so, p divides b]
Thus p is a common factor of a and b.
But this is a contradiction, since a and b have no common factor.
This contradiction arises by assuming √p a rational number.
Hence,√p is irrational.

Proof that √p is irrational if p is a prime number:

Assumption: Let's assume that √p is rational.

Definition of a rational number: A rational number is a number that can be expressed as the ratio of two integers.

Representation of √p as a ratio of two integers: If √p is a rational number, then it can be expressed as the ratio of two integers, say a and b, where b is not equal to 0.

Simplification of the ratio: We can simplify the ratio a/b to its lowest terms, which means that a and b have no common factors other than 1.

Representation of p as a square of a prime number: Since p is a prime number, it can be expressed as the square of a prime number, say p = q^2, where q is a prime number.

Substitution of p with q^2 in the ratio: Substituting p = q^2 in the ratio a/b, we get √p = √(q^2) = q(a/b).

q(a/b) is also a rational number: Since a/b is a rational number, q(a/b) is also a rational number.

Representation of q(a/b) as a ratio of two integers: We can represent q(a/b) as the ratio of two integers, say c and d.

c and d have no common factors other than 1: We can simplify the ratio c/d to its lowest terms, which means that c and d have no common factors other than 1.

Representation of q as a product of c and d: Since q = q*1 = q*(c/d) = (q*c)/d, we can represent q as the product of c and d, where c and d have no common factors other than 1.

Contradiction: This means that q has a common factor with a and b, which contradicts our assumption that a and b have no common factors other than 1.

Conclusion: Therefore, our assumption that √p is rational is false, and √p is irrational if p is a prime number.